#72252871

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃⁴ ⋊ M₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10620191
commutator, socle	C₂×C₃⁴	C₂×C₈
Fitting, prime radical	C₂²×C₃⁴	C₈
cosocle	C₃⋊(C₃⋊#622)	C₂²
hypercenter	C₂²	#10620191
hypocenter	C₃⁴	M₃₂
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

254 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^5,3^4|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^2g^2,f^c=g^2,f^d=f^2,g^a=fg^2,g^c=f,g^d=g^2,h^a=f^2g^2hi,h^c=f^2h^2,h^d=fgh,i^a=g^2hi^2,i^c=fgi^2,i^d=gi

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252878
maximal normal subgroup of	#72252878

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 3, F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.6^2*F.7^2*F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.6^2*F.8^2);;
gap> SetConjugate(coll, 8, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.7^2*F.8*F.9^2);;
gap> SetConjugate(coll, 9, 3, F.6*F.7*F.9^2);;
gap> SetConjugate(coll, 9, 4, F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);